The One Multiplication Fact You're Accidentally Teaching Twice

A student solves 3 x 4 without hesitation. Hand them 4 x 3 a minute later, and they treat it like a brand-new fact they've never seen.

Rather than treating every multiplication fact as its own separate thing to memorize, what if your students already know more facts than they think they do? This episode walks through a simple way to build that connection using a hands-on model students can see and touch.

In this episode, you'll explore:

• Why defining a property and actually using it as a strategy are two different things
• A simple instructional move that helps students see facts as related, not brand new
• Questions that help students make connections between facts

If you're re-teaching the same multiplication facts twice without realizing it, this episode will show you how to help your students see relationships instead.

Resources mentioned in this episode:

• Free Training: Why Your 3rd Graders STILL Don't Know Their Multiplication Facts
• Multiplication Facts Blueprint 

Want more support differentiating math instruction and running small groups with confidence? Check out my membership for third-grade teachers. →  Elementary Math Fam

If you liked this, follow the show so you never miss an episode and come visit me on Instagram @mrelementarymath.

Transcript

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Picture this. You're playing a multiplication game with your small group, and a student pulls a game card that asks them to solve three times four. They pause, count on their fingers, and land on twelve. The same student pulls another game card. This one asks them to solve four times three. They pause again, use the same finger counting, and get the same answer. And yes, technically, three times four and four times three are two different facts, but they are not strangers. They're connected. And today, I'm talking about how to teach the commutative property or turnaround facts from day one so your students stop relearning the same facts twice. Welcome to the Teaching Math Made Easy podcast. I'm your host Greg, otherwise known as Mr. Elementary Math. Here we share practical ideas to help your students learn and tips to help you feel more confident teaching elementary math. And we're back. Now let's get into it. You and I both know that 3 times 4 and 4 times 3 give the same answer every single time. Not sometimes, but every time. The technical name for this relationship is the commutative property. And if you've taught multiplication or addition, you know what I'm talking about, right? You've probably even said the words out loud in your classroom. And as educators, we need to make sure our students understand this as well. But here's the disconnect that can occur. Knowing the property exists and actually using it to cut down what facts your students have to know are two completely different things. Sometimes the commutative property gets taught as a rule or vocabulary word to know, not a strategy to use. It might be a lesson on properties of operations with a definition to record, and then instruction goes right back to treating every fact as its own separate island instead of being connected. So, what does teaching it as a strategy actually look like? Instead of introducing facts one at a time in a big, unconnected pile, you could group them by turnaround pairs from the very beginning. To build multiplication fact fluency, you could introduce 3x4 and 4x3 together in the same breath as two ways of asking the exact same question. Here's how this could sound in your classroom. You can have your students build an array with three groups of four. Then ask, what's the matching multiplication fact and the total? Your students might count each group of four or skip count by fours. Three groups in all to land on three times four equals twelve. Then you have them physically rotate the same array. Ask them, what matching multiplication fact do you have now? And what's the total? Hopefully you'll hear responses like this: four groups of three or four times three equals twelve. Then you ask them, how are these two arrays similar and how are they different? Guys, that's a very important question to ask, alright? And if you notice some students still need more support, you may need to show them the two arrays positioned side by side, alright? You want students to notice the total stays the same, although the groupings are different. By showing them how closely related one fact is to another, they already know you're showing them that they don't have to know a brand new fact, and that's the commutative property at work. Once students have that knowledge, continue to build your fact practice using the same strategy. Let's bring this all together with a few key takeaways. First, use arrays to explore the commutative property. Second, when a student confidently answers one fact, immediately ask them to answer the turnaround fact. For example, say a student nails six times seven without hesitation. Right after, ask them, so what's seven times six? Then watch to see what happens. You're looking for the light bulb moment when you hear, oh, it's the same. That's when you know they got it. If not, have them go back and build both arrays. Ask what changed and what did not. Most of the time, seeing them side by side makes the connection click because now the math is visible. In fact, I offer a fun commutative fact activity like this inside my course, the Multiplication Facts Blueprint. Here's why that light bulb moment matters more than it might seem on the surface. According to the researchers Jennifer Bay Williams and Gina Kling, real fluency is defined as someone being accurate, efficient, and flexible, not just fast. And flexibility means recognizing relationships between facts, like knowing that 7x4 and 4x7 are connected, instead of storing every fact as its own separate thing to know. When you teach turnaround pairs together, you're not only saving time, you're building exactly the kind of relational thinking that real fluency depends on. And most importantly, here's what it means for your students. Instead of learning 144 separate multiplication facts, students really only have to know about half that many, alright? Because every turnaround pair is one relationship, not two. Meaning, this is more than just a memorization shortcut, it's something bigger. When students see that 7 times 4 and 4 times 7 are connected, they're not just carrying less in their working memory, they're also starting to see multiplication as a system with relationships inside, instead of a giant list of unrelated facts to grind through. That's exactly what makes later math like division and algebra make more sense because they've already practiced looking for the relationship instead of just the answer. You got me? And for you, grouping facts by turnaround pairs is one of those rare, when I say rare, rare instructional tweaks that makes your job lighter, not heavier. Alright, guys? You're not implementing an unfamiliar strategy. You're simply teaching the same commutative property that you already teach. But this time, you're using it with more intention instead of just vocabulary words, students jot down and they completely forget. So this is one small shift in how you introduce and group facts for instruction. So give it a try. Introduce the two related facts together using arrays, ask a couple of questions, and watch for those light bulb moments. It's little to no risk with high reward. Being more intentional about how you use strategies to build fact fluency is the first step. Knowing which strategies your students need next is the next step. If multiplication facts have been lingering on your mind after listening to these episodes, I've got a free, yes edit, a free training for you called Why Your Third Graders Still Don't Know Their Multiplication Facts. And guys, it's not just for third grade students. If you have any students struggling with multiplication facts, this is for you. In it, we dig into why students get stuck even after weeks of practice and what real fluency actually looks like beyond speed and memorization. Like I said, it's free and you can watch it from wherever and whenever you want to, okay? So grab the link in the show notes. Thanks for joining me today. If you like this episode, follow the show on Apple Podcasts or Spotify so we can catch up next week. Until next time, keep teaching math with confidence because you've got this.